Some Romanians are planning to fly a balloon to the Moon. OK, to be more precise, what they’re planning is the use of a stratospheric balloon as a launch platform for their attempt to win Google’s Lunar X Prize. It’s not as crazy as it sounds… the disadvantages of launching from an unstable platform like a balloon may be more than offset by the advantages of launching from an altitude that is above most of the Earth’s atmosphere.
It looks like the Mythbusters tend to ignore air resistance.
In a recent episode, they claimed to have demonstrated that a horizontally fired bullet and a bullet that is simply dropped fall to the ground in the same amount of time. They were wrong. What they actually demonstrated is that air resistance causes the fired bullet to hit the ground more slowly.
Their argument would apply perfectly in a vacuum, as on the surface of the Moon, but not here on the Earth, where the bullet’s motion is governed not just by the laws of gravity, but also by the laws of a non-conservative force, namely air resistance. (Why is it non-conservative? Some of the bullet’s kinetic energy is converted into heat, as it travels through the air at high speed. Unless we also include the thermodynamics of the air into our equations of motion, the equations will not conserve energy, as the amount of kinetic energy converted into heat will just appear “lost”.)
The bullet’s velocity, v, can be written as v2 = vh2 + vv2, where vh is the horizontal and vv is the vertical component. The initial horizontal velocity is v0. The initial vertical velocity is 0.
Air resistance is proportional to the square of the bullet’s velocity. To be precise, acceleration due to air resistance will be
aair = κv2,
where κ is an unknown proportionality factor. (To be more precise, κ = ½cAρm, where c is the dimensionless drag coefficient, A is the bullet’s cross-sectional area, ρ is the density of the air, and m is the bullet’s mass.) The direction of the acceleration will be opposite the direction of the bullet’s motion.
The total acceleration of the bullet will have two components: a vertical component g due to gravity, and a component aair opposite the direction of the bullet’s motion.
The resulting equations of motion can be written as:
dvh/dt = –κvvh,
dvv/dt = g – κvvv.
Right here we can see the culprit: air resistance not only slows the bullet down horizontally, it also reduces its downward acceleration.
This is a simple system of two differential equations in the two unknown functions vh(t) and vv(t). Its solution is not that simple, unfortunately. However, it can be greatly simplified if we notice that given that vv << vh, v ≅ vh, and therefore, we get
dvh/dt = –κvh2,
dvv/dt = g – κvvvh.
This system is solved by
vh = 1/(κt + C1),
vv = vh[(½κt2 + C1t)g + C2].
Given
v0 = 1/(κt0 + C1)
we have
C1 = 1/(v0 – κt0),
which leads to
vh = v0/[κ(t – t0)v0 + 1],
or, if we set t0 = 0,
vh = v0/(κtv0 + 1),
Similarly, given vv(0) = 0, we get C2 = 0, thus
vv = gt(κv0t + 2)/(2κv0t + 2).
The distance traveled horizontally (sh) and vertically (sv) between t0 = 0 and t1 can be obtained by simple integration of the respective velocities with respect to t between 0 and t1:
sh = log(κv0t1 + 1)κ,
sv = g[κv0t1(κv0t1 + 2) – 2log(κv0t1 + 1)]/(2κv0)2.
The claim by the Mythbusters was that the time it took for the fired bullet to hit the ground was only ~40 ms more than the time it took for a dropped bullet to fall, which is a negligible difference. But it is not! Taking g ≅ 10 m/s2, it is easy to see that the time it takes for a bullet to fall from a height of 1 m, using the well-known formula ½gt2, is 447 ms; the difference measured by the Mythbusters is nearly 10% of this number!
Not only did the fired bullet take longer to hit the ground, the Myhtbusters’ exquisite setup allows us to calculate the bullet’s initial velocity v0 and drag coefficient κ. This is possible because the Mythbusters conveniently provided three pieces of information (I am using approximate numbers here): the length of the path that the bullet traveled sh ≅ 100 m), the height of the bullet at the time of firing (sv ≅ 1 m), and the time it took for the fired bullet to hit the ground. Actually, what they provided was the difference between the time for a fired vs. a dropped bullet to hit the ground, but we know what it is for the dropped bullet (and because it is never moving very rapidly, we can ignore air resistance in its case), so t1 = 447 + 40 = 487 ms. The solution is given by
κ = 0.0054 m–1,
v0 = 272.3 m/s.
Given a bullet cross-sectional area of A = 2 cm2 = 2 × 10–4 m2, an approximate air density of ρ = 1 kg/m3, and a bullet mass of m = 20 g = 0.02 kg, the dimensionless drag coefficient for the bullet can be calculated as c = 2κmAρ = 1.08, which is not at all unreasonable for a tumbling bullet. Of course the actual values of A and m may differ from the ones I’m using here, resulting in a different value for the dimensionless drag coefficient c.
It appears that the test launch of NASA’s Ares I-X rocket was successful. This is very good news indeed.
First, it was the multiverse. Then came Boltzmann brains. Now here’s another intriguing idea: the cancellation of the Superconducting Supercollider project in the 1990s and last year’s failure of one of the Large Hadron Collider’s magnets at CERN are just two manifestations of manifest bad luck brought about by the fact that the Higgs particle simply cannot be discovered; that its discovery at any time in the future propagates backwards in time, causing events that prevent its discovery in the first place.
An intriguing idea, though not precisely original, as something much like it was already published in the form of John Cramer‘s excellent science fiction novel, Einstein’s Bridge. And when I say intriguing, I mean intriguing… as the basis of a science fiction story. But as the basis of a scientific paper? Another adjective comes to my mind… appalling.
There are certain areas of life where decades of computer expertise are quite useless, and even a reasonably thorough knowledge of theoretical physics is only of marginal use. Replacing the rotted subfloor around a leaky toilet is one such area.
Yet this is what I am presently engaged in. So far so good… using some rather evil, foul-sounding power tools, I managed to cut out much of a square hole around the drainpipe, I’m only having trouble with some corners where the power tools don’t reach. Unfortunately, I found out that the subfloor in this bathroom is actually an inch thick, as opposed to the standard, 5/8″ board that I already bought… oh well, it wasn’t a big expense anyway, and perhaps I can use that board for some other purpose later on.
For now, it’s back to Home Depot to get a piece of inch-thick wood and also some advice on cutting out those nasty corners. Maybe they can suggest a method that would be slightly more efficient than the hammer-and-chisel approach which I attempted, with some limited success.
While I’m at it, I shall also inquire as to whether it is possible for them to cut my boards to shape to fit around the drainpipe, so that I wouldn’t have to attempt such precision cutting using my fairly limited skills and perhaps less-than-adequate set of tools. Not to mention that I value my fingers, and prefer to have all ten of them in the right place and in full working order after I’m done with all this…
But for now, it’s rest time. I have this nasty tensor algebra program to tackle, but no matter how difficult it is, I sweat a lot less doing it than when I’m cutting a subfloor with a circular saw.
If you’re a scientist or engineer, you don’t need to be a pacifist never to work for the military. J. Reece Roth, a 72-year old professor emeritus at the University of Tennessee, didn’t know this when he hired two graduate students (one from Iran, one from China) and when he took his laptop to China. His reward, for a lifetime of working hard and being a loyal citizen of the United States? Four years in prison.
I’m reading the autobiography of Fred Hoyle, and I’ve been perusing Wikipedia for background, in particular, reading about the Jodrell Bank radio telescope and its founder, Sir Bernard Lovell.
This is how I came across a news item from earlier this year, according to which Lovell recently revealed that back in 1963, he has been targeted by Soviet assassins during a visit to the Soviet Union.
This sounds improbable except… even in recent years, Russian intelligence agents/agencies have been using novel methods in assassination attempts (e.g., radioactive polonium in the case of Litvinenko). Further, the rationale Lovell gives is quite plausible: back in 1963, when satellite-based early warning systems were not yet available, something like Jodrell Bank may very well have served either as an over-the-horizon radar or perhaps using the Moon as a reflector.
Lovell promises to reveal more posthumously. What can I say? Our curiosity can wait. I wish him many more happy and healthy years.
I’m watching Cubers on the CBC, a documentary about the revival of interest in Rubik’s Cube, and a recent Rubik’s Cube solvers’ competition. What can I say… it takes me back.
I wouldn’t stand a chance competing in this crowd, but I did win the world’s first (as far as I know) Rubik’s Cube competition, held in Budapest in 1980. I completed my cube in 55 seconds, which wasn’t a very good time by my standards then (I often managed to solve the cube in well under 30 seconds) but it was enough to win.
These days, world class competitors solve the cube in 15 seconds or less. In addition to manual dexterity, such spectacular performances also require memorizing a large number of moves. And then I am not even going to mention the blindfold competitions, involving not just the “standard” 3×3×3 cube but the larger, 4×4×4 and 5×5×5 versions… such skills are hard to comprehend.
I can still solve my (3×3×3) cube without trouble (so long as I am allowed to use my eyes), but I only remember a relatively modest number of moves, which means that my solution is far from efficient. In other words… I am rusty. And my cube is sticky. Literally, it feels sticky on the outside (is the plastic decomposing?) and it’s a bit hard to turn. Still, on the third try, I managed to solve it in a minute an 45 seconds. Not bad, considering that I haven’t touched the thing in years.
Canadian astronaut Julie Payette is safely back on terra firma. During the post-mission press conference, she described the environment in which the International Space Station has been constructed, i.e., space, as “one of the most hostile”.
But… is it?
The funny thing is, while the human body is not designed to survive in space, if you suddenly found yourself floating outside the ISS, you’d have several seconds of useful consciousness before passing out. Further, if you managed to get back in before too many seconds have passed, you might survive the experience with only minor wear and tear and no permanent damage. (Yes, that infamous scene in 2001: A Space Odyssey is scientifically plausible.)
Compare this to the bottom of the ocean. How long would you survive under a pressure of several hundred atmospheres? Or consider the crater of an active volcano. How many seconds of consciousness would you enjoy before your body is vaporized?
And then I have not even considered something like the surface of the Sun (not to mention its interior.) Talk about hostile!
The thing is, space is hostile alright, but we are creatures of the near vacuum: we can briefly survive in vacuum, even return from it without major injury. Even for our machines, it’s much easier to survive in space than it is to survive elsewhere. Perhaps this should be seen as encouraging… once we master the challenge of getting out of the Earth’s gravity well economically, living in space may not be as hard as it sounds.
I am somewhat surprised that this idea has not become more popular yet, even though it’s yet the clearest “scientific proof” that we are, in fact, all immortal.
The “many worlds” interpretation of quantum mechanics says that the wave function never collapses: instead, every time a measurement is made, corresponding to each possible outcome a new universe comes into existence. You measure the spin of an electron and presto: there are now two universes, in one of which the spin is +1/2, in the other, -1/2. You flip a coin and presto: there are now two universes, the “heads”-universe and the “tails”-universe. (And many other universes in which the coin lands edgewise, explodes in mid-air, gets snatched by a passing eagle, or any other bizarre, improbable, but not impossible outcome that you can imagine.)
But if this is true, well, human death is just another measurement; and whereas in one universe, your heart might stop beating, in another, it beats one more. Or two more. Or two hundred million more.
In other words, as the universe keeps branching, you may cease to exist on many of those branches but there will always be branches on which you continue to live.
Think about it. That which you call your present consciousness will exist in an ever growing number of copies; some of those will be extinguished, but a few won’t be, not for a very, very, very long time. There is a continuous line from the here and now to the then and there, no matter how far that “then” is in the future, along which you continue to live. In other words, you can look forward to everlasting life… at least in a few of the many universes that await you.
How do you know if you’re on one of those “lucky” branches? Well, so long as you’re still alive, you are on a lucky branch, since the possibility exists that you will stay alive. Forever.
Of course there is a downside. Among the many parallel universes that represent possible futures, there are those in which you stay alive, but just barely, and in terrible pain and suffering. Or, you stay alive but you lose all your loved ones and even when you decide that it’s time to end your own life, you cannot… there is, after all, a nonvanishing probability that all your attempts at suicide fail.
But that doesn’t change the basic concept: in the multiverse, everyone is immortal. Although I am personally not too fond of the many worlds interpretation of quantum mechanics, I remain a little surprised that this idea has not yet become more popular among the religiously inclined.
Finch Avenue W.
This July has been the rainy season here in Ottawa. Indeed, we may yet break the all-time record for July rainfall. In some parts of Ottawa, homes and streets have been flooded, and yet we can consider ourselves lucky: unlike the folks in Toronto, we have not yet had to cope with a giant sinkhole in the middle of a major city road.
I was 6 years old 40 years ago today, visiting family in Romania with my parents. I did not really appreciate this moment (hey, I already read Jules Verne, isn’t going to the Moon a perfectly natural thing to do?) but I did see the first landing of a human being on another celestial body on television.
As Larry Kellogg said in his lunar-update newsletter, people who didn’t believe before that the Moon landing happened are unlikely to be convinced by this, but it’s still darn nice to be able to see nearly 40-year old footsteps on the Moon from orbit:
Apollo 14 footsteps
Forty years ago this morning, Apollo 11 was launched: Neil Armstrong and Buzz Aldrin were on their way to land at Mare Tranquilitatis, in the most significant journey in human history to this date.
The scary part is that this year also marks the 37th anniversary of the last trip to the Moon, indeed the last voyage by a human being beyond low Earth orbit.
I was only 6 when Armstrong and Aldrin landed on the Moon, and I had no doubt in my mind that by the time I turn 46, there would be people on the Moon, on Mars, possibly on select satellites of Jupiter and Saturn, perhaps even on their way to the stars.
Now that I am 46, I am doubtful that I will live long enough to see another human fly beyond low Earth orbit. This is not a pleasant thought. Perhaps I’ll be lucky enough to live another 40 years in good physical and mental health, and get a chance to be proven wrong.
Until then, I keep dwelling on the irony of the fact that nowadays, most of the documentaries you can find on manned deep space missions and exploration of the Moon are aired on the History Channel.
The premier Internet physics and astronomy preprint archive, ArXiv, seems to be having some serious problems tonight. I used the catchup interface to check for new papers, only to find messages like this:
Apparently all new papers are unavailable, and many older papers, too… I checked briefly and found papers dating back to last October that appear to have vanished. Including some half a dozen or so papers of my own.
I sure hope they keep backups!
I recently read a review of Weinberg’s wonderful new book, Cosmology, a 2009 sequel of sorts to his 1972 classic, Gravitation and Cosmology. The reviewer mentioned two other books, that of Mukhanov and that of Dodelson, as books worth having. Mukhanov’s Physical Foundations of Cosmology was already on my bookshelf (and, like the reviewer, I also consider it worth having) but not Dodelson’s book… so I decided to buy it.
I was not disappointed: it is an excellent cosmology book. In particular, it offers a very thorough introduction to the quantitative aspects of physical cosmology.
However… although the book was published only six years ago, it feels surprisingly dated. Through no fault of the author, to be clear: it’s just that cosmology has made tremendous progress in a few short years. I can think of two things in particular: results from the Wilkinson Microwave Anisotropy Probe (WMAP), first released in 2005, providing precision maps of the cosmic microwave background, allowing accurate detection of the so-called acoustic peaks; and ever improving large scale galaxy surveys, notably the Sloan Digital Sky Survey (SDSS), providing spectra for many hundreds of thousands of galaxies, yielding 3D density maps of the deep cosmos that can be used to test models of structure formation.
The results speak for themselves. For instance, Dodelson’s book gives 12.6 ± 1.1 billion years as the age of the Universe… in contrast, the latest WMAP result is 13.73 ± 0.12 billion years, a tenfold improvement in the accuracy of the estimate. I guess it’s not an enviable task to write a book for a field that is changing as rapidly as Modern Cosmology… which also happens to be the title of Dodelson’s book.
I found this gem of a sentence on the Web site of the Embassy of the Islamic Republic of Iran here in Ottawa:
“The Islamic Republic of Iran will register acts of all these states whose records are filled with support for terrorism, pro-colonialist policies for colonizing the oppressed nations, support for the despotic regimes, arming some with weapons of mass destruction and support for anarchy in all parts of the world as disdainful behavior and stipulates that they can not cast doubt on the excellent democratic election held recently in the Islamic Republic of Iran by no means advising them to change their miscalculated approach vis-à-vis the developments having taken place in Iran because designers of the chess game are closely monitoring their behavior and calculating them in the future relations.”
This sentence reminds me of Soviet-era propaganda leaflets. I wonder if the Islamic Republic of Iran has hired propagandists from the former Soviet Union who were left unemployed after 1991.
Anyhow, what exactly are they saying here? Something is wrong with this sentence. They say that,
The Islamic Republic of Iran
- will register, as disdainful behavior,
- acts of all these states whose records are filled with
- support for terrorism,
- pro-colonialist policies for colonizing the oppressed nations,
- support for the despotic regimes, arming some with weapons of mass destruction and
- support for anarchy in all parts of the world
and
- stipulates that they can not cast doubt on the excellent democratic election held recently in the Islamic Republic of Iran
by no means advising them to change their miscalculated approach vis-à-vis the developments having taken place in Iran
because
designers of the chess game are closely monitoring their behavior and calculating them in the future relations.
Hmmm… they seem to be telling us that despite all the bad things they say about our disdainful behavior, they are NOT advising us to change our miscalculated approach. The reason for this surprising advice has to do with the designers of the game of chess. Okay, I know that chess may have arrived in Europe from India by way of Persia, but what do the long dead inventors of one of the world’s most popular games have to do with the reelection of Ahmedinejad?
Maybe they are trying to confuse us intentionally, in order to deflect our attention away from a study that suggests that the election was seriously rigged. They really shouldn’t bother. This study says that the election was likely rigged because the final two digits of provincial results show unlikely statistics. But unlikely is not the same as impossible, and unless they can quantify how much more likely this outcome is in a rigged election, the study means nothing; after all, 1-2-3-4-5-6 is as likely to win in a random 6/49 lottery draw as any other number combination, and if they pick these numbers next week, it does not prove fraud by the lottery corporation. For that claim, one would also have to quantify the increased likelihood that a fraudulent draw is more likely to produce the 1-2-3-4-5-6 result when compared to a truly random draw.
I was channel-surfing for news this morning, and I caught a segment on CTV’s morning show about “dirty electricity”.
I shall refrain from calling the gentleman being interviewed using a variety of unflattering names, because it would not be polite, and in any case, it’s not the person but the message that I take issue with.
Basically, he put a bunch of electronic devices like cordless phones, baby monitors, Wi-Fi routers or even fluorescent light bulbs on a test bench, plugged them in, and then held a contraption with an antenna and a speaker close to them. The contraption was making loud noises, from which this gentleman concluded that these devices “emit radiation”, and “send dirty electricity back through the wires”.
So then… what? The whole Universe is emitting similar radiation at radio frequencies. Any warm object, including the walls of your house, emits radiation at such frequencies and higher. And why should I care?
Of course, it helps dropping a few scary phrases like, “skyrocketing rates of autism”. Oh, he wasn’t saying that they are related. Why should he? Merely mentioning autism while he’s talking about “dirty electricity” is enough to suggest a connection.
Just to be clear about it, almost all electronic devices emit radio frequency radiation that can then be picked up by a suitable receiver and converted into loud and scary noise. When I was 10 or so and got my first pocket calculator, I had endless fun holding it close to an AM receiver and listening to its “song”. Later, when I had my first programmable calculator, I could tell by listening to the sounds on a nearby radio if it was still executing a program, or even if it displayed the expected result or just showed an error condition. Modern calculators use so little power that their transmissions cannot be picked up so easily, but does this mean that the old calculators were a health threat? Of course not.
At such low frequencies, electromagnetic radiation does not interact with our bodies in harmful ways. To cause genetic damage, for instance, much shorter wavelengths would be needed, you need to go at least to the ultraviolet range to produce ionization and, possibly, damage to DNA. At lower frequencies, most emissions are not even absorbed by the body very effectively. The little energy that is being absorbed may turn into tiny currents, but those are far too tiny to have any appreciable biological impact. Note that we are not talking about holding a cell phone with a, say, 0.3W transmitter just an inch from your brain (though even that, I think, is probably quite harmless, never mind sensationalist claims to the contrary); we are talking about a few milliwatts of stray radio frequency emissions not mere inches, but feet or more from a person.
As to “dirty electricity”, any device that produces a capacitive or inductive load on the house wiring will invariably feed some high frequency noise back through the wiring. Motors are the worst offenders, like vacuum cleaners or washing machines. Is this a problem? I doubt it. House wiring already acts as a powerful transmission antenna, continuously emitting electromagnetic waves at 60 Hz (in North America); so what if this emission is modulated further by some higher frequency noise?
But even if I am wrong about all of this, and low-frequency, low-energy electromagnetic radiation has a biological effect after all… study it by all means, yes, but it is no excuse for CTV to bring a scaremongerer with his noisy gadget (designed clearly with the intent to impress, not measure) on live television.
The reason why I am concerning myself with more Maxima examples for relativity is that I am learning some subtle things about Brans-Dicke theory and the Parameterized Post-Newtonian (PPN) formalism.
Brans-Dicke theory is perhaps the simplest modification of general relativity. Instead of the gravitational constant, G, the theory has a scalar field φ, and the theory’s Lagrangian now reads
L = [φR − ω∂μφ∂μφ/φ] / 16π.
Here, R is the curvature scalar and ω is an unspecified constant of the theory.
The resulting field equations are just like Einstein’s, except for two things. First, the field equations for the metric now have additional terms containing derivatives of φ; second, there is a new field equation for the scalar field φ that basically says that the d’Alembertian of φ is proportional to the trace of the stress-energy tensor.
Clever people tell you that Brans-Dicke theory is practically excluded by solar system data, as it would only work for insanely high values of ω. They demonstrate this by building approximate solutions for the theory using the PPN formalism, and find that one of the PPN parameters, γ, will have the value of γ = (1 + ω) / (2 + ω); on the other hand, observations by the Cassini spacecraft restrict γ to |γ − 1| < 2.3 × 10−5, so |ω| must be at least 40,000.
Now here’s the puzzling bit: if you solve Brans-Dicke theory in a vacuum, you find that the celebrated Schwarzschild solution of general relativity still applies: keeping φ constant, you just get back this common solution which is known to fit solar system data well, and which has, most importantly, γ = 1 and the value of ω doesn’t matter.
So which is it? Is it γ = 1 or is it γ = (1 + ω) / (2 + ω)? Something is amiss here.
This dilemma can be resolved once you realize that whereas general relativity has a unique spherically symmetric, static vacuum solution, this is not the case for Brans-Dicke theory. This theory has an infinite family of spherically symmetric, static vacuum solutions. Indeed, I think you could actually use the value of γ to parameterize this solution space. However, once you allow some matter into that vacuum, no matter how little, you are locked in to a specific solution, for which γ = (1 + ω) / (2 + ω). In other words, the only vacuum solution that is consistent with the notion of taking the limit of a matter solution by gradually removing matter is NOT the Schwarzschild solution of general relativity, but another, incompatible solution.
This has extremely important implications for our work on MOG. So far, we have obtained a vacuum solution that appears consistent with observations on scales from the solar system to cosmology. However, a recent paper by Deng et al. challenges this work by suggesting that the MOG PPN parameter γ is not 1 and hence, the theory runs into the same trouble as Brans-Dicke theory in the solar system. Is this true? Did we pick a vacuum solution that happens to be inconsistent with matter solutions? This is what I am trying to investigate.
Some moderately interesting Maxima examples.
First, this is how we can prove that the covariant derivative of the metric vanishes (but only if the metric is symmetric!)
load(itensor); imetric(g); ishow(covdiff(g([],[i,j]),k))$ %,ichr2$ ishow(contract(canform(contract(canform(rename(expand(%)))))))$ ishow(covdiff(g([i,j],[]),k))$ %,ichr2$ ishow(canform(contract(rename(expand(%)))))$ decsym(g,2,0,[sym(all)],[]); decsym(g,0,2,[],[sym(all)]); ishow(covdiff(g([],[i,j]),k))$ %,ichr2$ ishow(contract(canform(contract(canform(rename(expand(%)))))))$ ishow(covdiff(g([i,j],[]),k))$ %,ichr2$ ishow(canform(contract(rename(expand(%)))))$
Next, the equation of motion for a perfect fluid:
load(itensor);
imetric(g);
decsym(g,2,0,[sym(all)],[]);
decsym(g,0,2,[],[sym(all)]);
defcon(v,v,u);
components(u([],[]),1);
components(T([],[i,j]),(rho([],[])+p([],[]))*v([],[i])*v([],[j])
-p([],[])*g([],[i,j]));
ishow(covdiff(T([],[i,j]),i))$
ishow(canform(%))$
ishow(canform(rename(contract(expand(%)))))$
%,ichr2$
canform(%)$
ishow(canform(rename(contract(expand(%)))))$
Finally, the equation of motion in the spherically symmetric, static case:
load(ctensor); load(itensor); K:J([i],[])=covdiff(T([i],[j]),j); E:ic_convert(K); ct_coords:[t,r,u,v]; lg:ident(4); lg[1,1]:B; lg[2,2]:-A; lg[3,3]:-r^2; lg[4,4]:-r^2*sin(u)^2; depends([A,B,T,rho,p],[r]); derivabbrev:true; cmetric(); christof(mcs); J:[0,0,0,0]; ev(E); T:ident(4); T[1,1]:rho; T[2,2]:T[3,3]:T[4,4]:p; J,ev;
These examples are probably not profound enough to include with Maxima, but are useful to remember.